Chiral Bag Research Articles

Edge states and the η invariant

We propose a relation between the η invariant on a manifold with boundary, the η invariants of edge states, and the η invariant in an infinite volume limit. With the example of planar fermions with bag and chiral bag boundary conditions we show that this relation holds whenever edge states are sufficiently well-localized near the boundary. As a by-product we show that the spectrum of edge modes for chiral bag boundary conditions is linear but bounded.

Conformal Dirac–Einstein equations on manifolds with boundary.

In this paper we study Dirac–Einstein equations on manifolds with boundary, restricted to a conformal class with constant boundary volume, under chiral bag boundary conditions for the Dirac operator. We characterize the bubbling phenomenon, also classifying ground state bubbles. Finally, we prove an Aubin-type inequality and a related existence result.

Heat kernel, spectral functions and anomalies in Weyl semimetals

Being motivated by applications to the physics of Weyl semimetals we study spectral geometry of Dirac operator with an abelian gauge field and an axial vector field. We impose chiral bag boundary conditions with variable chiral phase θ on the fermions. We establish main properties of the spectral functions which ensure applicability of the ζ function regularization and of the usual heat kernel formulae for chiral and parity anomalies. We develop computational methods, including a perturbation expansion for the heat kernel. We show that the terms in both anomalies which include electromagnetic potential are independent of θ.

Historical Introduction to Chiral Quark Models

Chiral symmetry, and its dynamical breaking, has become a cornerstone in the description of the hadron’s phenomenology at low energy. The present manuscript gives a historical survey on how the quark model of hadrons has been implemented along the last decades trying to incorporate, among other important non-perturbative features of quantum chromodynamics (QCD), the dynamical chiral symmetry breaking mechanism. This effort has delivered different models such as the chiral bag model, the cloudy bag model, the chiral quark model or the chiral constituent quark model. Our main aim herein is to provide a brief introduction of the Special Issue “Advances in Chiral Quark Models” in Symmetry and contribute to the clarification of the differences among the above-mentioned models that include the adjective chiral in their nomenclature.

Weyl anomaly induced Fermi condensation and holography

Recently it is found that, due to Weyl anomaly, a background scalar field induces a non-trivial Fermi condensation for theories with Yukawa couplings. For simplicity, the paper consider only scalar type Yukawa coupling and, in the BCFT case, only for a specific boundary condition. In these cases, the Weyl anomaly takes on a simple special form. In this paper, we generalize the results to more general situations. First, we obtain general expressions of Weyl anomaly due to a background scalar and pseudo scalar field in general 4d BCFTs. Then, we derive the general form of Fermi condensation from the Weyl anomaly. It is remarkable that, in general, Fermi condensation is non-zero even if there was not a non-vanishing scalar field background. Finally, we verify our results with free BCFT with Yukawa coupling to scalar and pseudo-scalar background potential with general chiral bag boundary condition and with holographic BCFT. In particular, we obtain the shape and curvature dependence of the Fermi condensate from the holographic one point function.

Baryon as a Quantum Hall Droplet and the Quark-Hadron Duality.

We show that the recent proposal to describe the N_{f}=1 baryon in the large number of the color limit as a quantum Hall droplet can be understood as a chiral bag in a (1+2)-dimensional strip using the Cheshire Cat principle. For a small bag radius, the bag reduces to a vortex line which is the smile of the cat with flowing gapless quarks all spinning in the same direction. The disk enclosed by the smile is described by a topological field theory due to the Callan-Harvey anomaly outflow. The chiral bag naturally carries the unit baryon number and spin 1/2N_{c}. The generalization to arbitrary N_{f} is discussed.

Fractional fermion number and Hall conductivity of domain walls

In this letter the fractional fermion number of thick domain walls is computed. The analysis is achieved by developing the heat kernel expansion of the spectral eta function of the Dirac Hamiltonian governing the fermionic fluctuations around the domain wall. A formula is derived showing that a non null fermion number is always accompanied by a Hall conductivity induced on the wall. In the limit of thin and impenetrable walls the chiral bag boundary conditions arise, and the Hall conductivity is computed for this case as well.

5a-A-5 g_A Problem in the Chiral Bag Model

5a-A-5 g_A Problem in the Chiral Bag Model

Edge states at phase boundaries and their stability

We analyze the effects of Robin-like boundary conditions on different quantum field theories of spin 0, 1/2 and 1 on manifolds with boundaries. In particular, we show that these conditions often lead to the appearance of edge states. These states play a significant role in physical phenomena like quantum Hall effect and topological insulators. We prove in a rigorous way the existence of spectral lower bounds on the kinetic term of different Hamiltonians, even in the case of Abelian gauge fields where it is a non-elliptic differential operator. This guarantees the stability and consistency of massive field theories with masses larger than the lower bound of the kinetic term. Moreover, we find an upper bound for the deepest edge state. In the case of Abelian gauge theories, we analyze a generalization of Robin boundary conditions. For Dirac fermions, we analyze the cases of Atiyah–Patodi–Singer and chiral bag boundary conditions. The explicit dependence of the bounds on the boundary conditions and the size of the system is derived under general assumptions.

Band rearrangement through the 2D-Dirac equation: Comparing the APS and the chiral bag boundary conditions

The Dirac equation on a two-disk is studied under the chiral bag boundary condition, where the mass is treated as a parameter ranging over all real numbers. The eigenvalues as functions of the parameter are compared with those obtained under the APS boundary condition studied in a previous paper of authors (Iwai and Zhilinskii, 2015). Discrete symmetry (or pseudo-symmetry) of the boundary condition as well as the Hamiltonian is studied to explain the difference between the patterns of eigenvalues under the chiral bag and the APS boundary conditions. The spectral flow for a one-parameter family of operators is the net number of eigenvalues passing through zero in the positive direction as the parameter runs. It was demonstrated in the previous paper that the spectral flow is useful to understand the characteristic of eigenvalue pattern of the Dirac equation with the APS boundary condition. However, to capture the feature of eigenvalue pattern under the chiral bag boundary condition, one needs to introduce an extended notion of spectral flow. The eigenvalue patterns under the both boundary conditions are compared with a semi-quantum description of energy-band rearrangement.

A dynamical mass and confinement in nucleon model

A possible dynamical softening of the Chiral bag is studied in connection with the chiral quark model and the Nambu–Jona–Lasinio (NJL) model. It is demonstrated that the bag functions constructed by the pion degrees is able to confine quark through generation of the rising effective mass of valence quark due to the pion dressing and break chiral symmetry dynamically. The tendency of the bag functions toward the MIT bag is illustrated in the large-n limit for a solitonic configuration of the chiral field Un(r) for which U1(r) is the Skyrmion profile. The mass running with the length scale agrees with that of the NJL Model for n ≥ 2.

Edge states: topological insulators, superconductors and QCD chiral bags

The dynamics of the magnetic field in a superconducting phase is described by an effective massive bosonic field theory. If the superconductor is confined in a domain M with boundary \partial M, the boundary conditions of the electromagnetic fields are predetermined by physics. They are time-reversal and also parity invariant for adapted geometry. They lead to edge excitations while in comparison, the bulk energies have a large gap. A similar phenomenon occurs for topological insulators where appropriate boundary conditions for the Dirac Hamiltonian also lead to similar edge states and an 'incompressible bulk'. They give spin-momentum locking as well. In addition time-reversal and parity invariance emerge for adapted geometry. Similar edge states appear in QCD bag models with chiral boundary conditions.

Recent Status of the Chiral Bag Model

In this note, recent status of the chiral bag model is presented. As it combines the MIT quark bag model and the Skyrme model, the chiral bag model interpolates the two models smoothly as a function of the chiral bag radius R. The correct limit of R→∞ is reproduced by including the higher order terms in the Ω expansion of the cranking method. It resolves the so-called small gA problem in a class of models where the semiclassical method is used.

Nonperturbative proton stability

Proton decay is a generic prediction of GUT models and is therefore an important channel to detect the existence of unification or to set limits on GUT models. Current bounds on the proton lifetime are around ${10}^{33}\text{ }\text{ }\mathrm{years}$, which sets stringent limits on the GUT scale. These limits are obtained under ``reasonable'' assumptions about the size of the hadronic matrix elements. In this paper we present a nonperturbative calculation of the hadronic matrix elements within the chiral bag model of the proton. We argue that there is an exponential suppression of the matrix elements, due to nonperturbative QCD, that stifles proton decay by orders of magnitude---potentially $\mathcal{O}({10}^{\ensuremath{-}10})$. This suppression is present for small quark masses and is due to the chiral symmetry breaking of QCD. Such a suppression has clear implications for GUT models and could resuscitate several scenarios.

Spectral Action for Torsion with and without Boundaries

We derive a commutative spectral triple and study the spectral action for a rather general geometric setting which includes the (skew-symmetric) torsion and the chiral bag conditions on the boundary. The spectral action splits into bulk and boundary parts. In the bulk, we clarify certain issues of the previous calculations, show that many terms in fact cancel out, and demonstrate that this cancellation is a result of the chiral symmetry of spectral action. On the boundary, we calculate several leading terms in the expansion of spectral action in four dimensions for vanishing chiral parameter $\theta$ of the boundary conditions, and show that $\theta=0$ is a critical point of the action in any dimension and at all orders of the expansion.

Green functions for the Dirac operator under local boundary conditions and applications

In this article, we define the Green function for the Dirac operator under two local boundary conditions: the condition associated with a chirality operator (also called the chiral bag boundary condition) and the MIT bag boundary condition. Then we give some applications of these constructions for each Green function. From the existence of the chiral Green function, we derive an inequality on a spin conformal invariant which, in some cases, solves the Yamabe problem on manifolds with boundary. Finally, using the MIT Green function, we give a simple proof of a positive mass theorem previously proved by Escobar.

NUCLEON IN NUCLEAR MEDIUM

The behavior of a nucleon in nuclear medium is discussed in Chiral Color Dielectric Model. It is assumed that the nucleons in nuclear medium produces a background dielectric field and the quark and dielectric field equations are solved self consistantly in presence of the dielectric field. A nucleon in nuclear medium is then constructed by means of standard procedure followed in chiral bag models. The corrections due to center of mass motion, color magnetic interaction and meson interaction are included. The calculations show that the nucleon becomes bigger in the medium but its mass does not change much. It is found that beyond a certian density, bound solutions in which quarks are bound in self-generated dielectric field are not possible. Thus, the calculations indicate that there is a critical density beyond which the matter consists of deconfined quarks.

Electromagnetic baryon form factors from holographic QCD

In the holographic model of QCD suggested by Sakai and Sugimoto, baryons are chiral solitons sourced by D4 instantons in bulk of size 1/\sqrt{\lambda} with \lambda=g^2N_c. We quantize the D4 instanton semiclassically using \hbar=1/(N_c\lambda) and non-rigid constraints on the vector mesons. The holographic baryon is a small chiral bag in the holographic direction with a Cheshire cat smile. The vector-baryon interactions occur at the core boundary of the instanton in D4. They are strong and of order 1/\sqrt{\hbar}. To order \hbar^0 the electromagnetic current is entirely encoded on the core boundary and vector-meson dominated. To this order, the electromagnetic charge radius is of order \lambda^0. The meson contribution to the baryon magnetic moments sums identically to the core contribution. The proton and neutron magnetic moment are tied by a model independent relation similar to the one observed in the Skyrme model.

On a spin conformal invariant on manifolds with boundary

On a n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric, a spin structure and a chirality operator, we study some properties of a spin conformal invariant defined from the first eigenvalue of the Dirac operator under the chiral bag boundary condition. More precisely, we show that we can derive a spinorial analogue of Aubin's inequality.

Quark condensates in the chiral bag with the Nambu–Jona-Lasinio interaction

We discuss the quark condensate of the vacuum inside the baryon. We analyze the $1+1$-dimensional chiral bag in analogy with the realistic $1+3$-dimensional one. The Nambu\char21{}Jona-Lasinio (NJL) type interaction is used to investigate the quark condensate in the chiral bag. Considering the strong meson-quark coupling, we solve the mean field solution to the scalar and pseudoscalar channels. Extracting the finite values of the chiral Casimir energy and the quark condensate by proper regularization, the resulting self-consistent equation allows a generation of a finite dynamical quark mass inside the bag.